Appearance
3.6 — Real Matter: Phase Changes, Real Gases and the Third Law
The ideal gas of Chapter 3.2 can never become a liquid. That is not a limitation of the equations; it is built into the assumptions. Molecules with no size and no attraction have no reason ever to stick together, so no matter how hard you squeeze or how cold you go, PV = nRT describes a gas and only a gas, all the way down to absolute zero and all the way up to infinite pressure.
Real matter condenses, freezes, and does several stranger things besides. This chapter puts back the two assumptions that were dropped and follows the consequences: an equation of state that predicts liquefaction, a temperature above which a gas cannot be liquefied at any pressure, a diagram that says which phase you get under any conditions, and finally a law about absolute zero that explains why the entropy curve at the end of Chapter 3.5 had to start where it did.
What actually happens when you compress a gas
Take carbon dioxide in a cylinder with a piston, hold it at a fixed temperature, and squeeze it slowly while recording pressure against volume. Do it at several temperatures.
These curves are real measurements, made by Thomas Andrews in 1869, and they show three distinct behaviours.
At high temperature (the top curves) the gas behaves almost exactly as PV = nRT predicts: squeeze the volume in half and the pressure doubles. Nothing dramatic happens at any pressure.
At low temperature (the bottom curves) the story changes. Squeeze, and the pressure rises normally for a while. Then the curve goes flat. You keep pushing the piston in and the pressure does not rise at all. Look through the cylinder wall at this point and you will see droplets of liquid CO₂ forming. The gas is condensing, and the flat section is the region where liquid and vapour coexist. The volume keeps shrinking because gas is turning into liquid, which occupies far less room, and the pressure stays pinned at whatever the vapour pressure is at that temperature. Only when the last of the gas has condensed does the curve turn sharply upwards — and it turns very sharply indeed, because a liquid is nearly incompressible.
In between the flat section gets shorter as the temperature rises, until at one particular temperature it shrinks to a single point. That is the critical point, and above it there is no flat section at all: no matter how hard you squeeze, the gas never separates into two visible phases.
For carbon dioxide the critical temperature is 31.0 °C and the critical pressure is 73.8 atmospheres. Above 31 °C, carbon dioxide cannot be liquefied by pressure alone, which is a striking thing for a substance to have a temperature limit on. For water the critical point is at 374 °C and 218 atm; for helium it is at 5.2 K, which is why helium was the last gas anyone managed to liquefy.
What is happening at the critical point
Approach the critical point from below and something visible happens. The liquid and the gas become more and more alike — the liquid expands and thins as it is heated, the vapour is compressed and thickens — until at the critical point their densities become equal and the boundary between them, the meniscus, simply vanishes. There is no longer any difference between the two phases to have a boundary between.
Just at that point the substance scatters light strongly and turns milky, an effect called critical opalescence. The cause is that density fluctuations, which are normally microscopic, grow to the size of the wavelength of light because it costs almost no energy to convert a patch of liquid into a patch of gas. Einstein explained it quantitatively in 1910.
Beyond the critical point the material is a supercritical fluid: it fills its container like a gas, dissolves things like a liquid, and has no surface. Supercritical CO₂ is used industrially to decaffeinate coffee and to extract hops for beer, because it dissolves the target compounds, penetrates solids as easily as a gas, and then evaporates completely away when the pressure is released, leaving no solvent residue.
The van der Waals equation, derived
Johannes Diderik van der Waals set out in his 1873 doctoral thesis to fix the ideal gas law with two corrections, one for each dropped assumption. He got the Nobel Prize for it in 1910.
Correction one: molecules take up room. The ideal law says the gas can be compressed to zero volume. It cannot — the molecules themselves occupy space, and the volume actually available for them to move in is less than the container volume. So replace V with V - nb, where b is the excluded volume per mole, a property of the substance. For CO₂, b = 4.27\times10^{-5} m³/mol, roughly 43 cm³ per mole, which is the volume of about 6\times10^{23} molecules packed together.
Correction two: molecules attract each other. A molecule deep in the gas is pulled equally in all directions and feels no net force. A molecule about to strike the wall is different: it has gas behind it and wall in front, so the pull of its neighbours is entirely backwards, and it hits the wall a little more gently than it otherwise would. So the measured pressure is lower than the ideal-gas pressure.
How much lower? The reduction depends on two things, both proportional to density: how many molecules are arriving at the wall, and how many neighbours each has pulling it back. Two factors of n/V means the correction goes as (n/V)^2:
P_{\text{measured}} = P_{\text{ideal}} - a\left(\frac{n}{V}\right)^2
where a measures the strength of the attraction. So P_{\text{ideal}} = P + a n^2/V^2. Putting both corrections into PV = nRT:
\boxed{\left(P + \frac{an^2}{V^2}\right)(V - nb) = nRT}
Read aloud: P plus a n-squared over V-squared, times V minus n b, equals n R T. The first bracket is the pressure the molecules would exert if they did not attract each other; the second is the volume they actually have to move in.
For carbon dioxide, a = 0.364 Pa m⁶/mol² and b = 4.27\times10^{-5} m³/mol. The two constants are fitted per substance, which is the equation's honest weakness — it is a physically motivated fit, not a derivation from first principles.
It predicts the critical point
Here is what makes the equation more than a curve-fitting exercise. Plot van der Waals isotherms and at high temperature they look like ideal gas curves. Lower the temperature and they develop a wiggle: a section where the curve goes down then up then down again. Lower still and the wiggle grows.
The critical point is where the wiggle first appears — the temperature at which the curve has a horizontal inflection, meaning both the slope and the curvature vanish:
\left(\frac{\partial P}{\partial V}\right)_T = 0 \quad\text{and}\quad \left(\frac{\partial^2 P}{\partial V^2}\right)_T = 0
The first says the curve is momentarily flat; the second says it is not curving either way at that spot. Together they pick out the exact place where a dip and a rise are just about to be born.
Solve the van der Waals equation for P:
P = \frac{nRT}{V-nb} - \frac{an^2}{V^2}
Work per mole to keep it clean (n = 1), differentiate once:
\frac{dP}{dV} = -\frac{RT}{(V-b)^2} + \frac{2a}{V^3} = 0
and again:
\frac{d^2P}{dV^2} = \frac{2RT}{(V-b)^3} - \frac{6a}{V^4} = 0
From the first: RT = 2a(V-b)^2/V^3. From the second: RT = 3a(V-b)^3/V^4. Set them equal:
\frac{2a(V-b)^2}{V^3} = \frac{3a(V-b)^3}{V^4}
Cancel a and (V-b)^2, and multiply both sides by V^4:
2V = 3(V-b) \quad\Longrightarrow\quad 2V = 3V - 3b \quad\Longrightarrow\quad V_c = 3b
Substitute back into RT = 2a(V-b)^2/V^3 with V = 3b:
RT_c = \frac{2a(2b)^2}{27b^3} = \frac{8a}{27b} \quad\Longrightarrow\quad \boxed{T_c = \frac{8a}{27Rb}}
And the critical pressure, from the equation of state at (V_c, T_c):
P_c = \frac{R T_c}{3b - b} - \frac{a}{9b^2} = \frac{8a}{27b\cdot 2b} - \frac{a}{9b^2} = \frac{4a}{27b^2} - \frac{3a}{27b^2} = \boxed{\frac{a}{27b^2}}
Check it against carbon dioxide.
T_c = \frac{8(0.364)}{27(8.314)(4.27\times10^{-5})} = \frac{2.912}{9.585\times10^{-3}} = 304\ \text{K} = 31\ ^\circ\text{C}
P_c = \frac{0.364}{27(4.27\times10^{-5})^2} = \frac{0.364}{4.923\times10^{-8}} = 7.39\times10^{6}\ \text{Pa} = 73\ \text{atm}
Measured: 31.0 °C and 73.8 atm. The equation predicted a critical point from two constants fitted at ordinary conditions, and got both numbers right. That is why van der Waals's thesis mattered — it showed the liquid and gas states are not two different kinds of thing needing two different theories, but one substance described by one equation.
Where it still fails
The wiggle below the critical temperature has a section where dP/dV is positive: squeeze the fluid and the pressure drops. That is mechanically unstable and nothing behaves that way. Reality replaces the wiggle with the flat coexistence line, positioned by Maxwell's equal-area construction — draw the horizontal line at the height that makes the two lobes of the wiggle equal in area.
The parts of the wiggle just either side of the flat line are not entirely fictional, though. They correspond to metastable states which really can be prepared if you are careful: superheated liquid (above its boiling point but not boiling, for want of a nucleation site) and supercooled vapour. Superheated water is why a mug of water heated in a microwave can sit still at 105 °C and then erupt violently when a spoon is dipped in, giving the bubbles somewhere to start.
Phase diagrams
Plot pressure against temperature and mark which phase is stable in each region. The result is a map of a substance's entire behaviour.
Read the map carefully, because every feature means something.
The three lines are where two phases coexist. The solid–liquid line is the melting curve, the liquid–gas line is the boiling curve, and the solid–gas line is the sublimation curve. Crossing a line is a phase change, and it absorbs or releases the latent heat of Chapter 3.1.
The triple point is where all three lines meet, and it is a single unique combination of pressure and temperature at which solid, liquid and gas coexist in equilibrium. For water it is 0.01 °C at 611.657 Pa — about 0.006 atmospheres. It is so precisely reproducible that from 1954 until 2019 the kelvin was defined by it: the triple point of water was declared to be exactly 273.16 K. The 2019 redefinition moved to fixing k_B instead, but the triple point remains the standard practical fixed point.
The critical point ends the liquid–gas line. Notice it ends — it does not continue to infinity. That is the meaning of the last section: above the critical temperature there is no distinction between liquid and gas, so there is no boundary to draw. You can go around the end of the line, from liquid to gas, without ever crossing a boundary and without ever seeing anything boil.
By contrast the solid–liquid line has no critical point and continues upward indefinitely as far as anyone has measured. A solid and a liquid differ in symmetry — a crystal has a repeating lattice, a liquid does not — and a symmetry is either present or absent, with no continuous way to blur one into the other.
Water's backwards melting line
On nearly every phase diagram the solid–liquid line leans slightly right: raise the pressure and the melting point goes up, because squeezing favours the denser phase and the solid is denser. Water's leans left. Raise the pressure on ice at −1 °C and it melts.
The reason is the density anomaly from Chapter 3.1: ice is less dense than liquid water, so squeezing favours the liquid. The relationship is exact, and it is called the Clausius–Clapeyron equation:
\frac{dP}{dT} = \frac{L}{T\,\Delta V}
which reads: the slope of a phase boundary equals the latent heat divided by the temperature times the volume change across the boundary. For melting ice, \Delta V is negative — the volume shrinks on melting — so the slope is negative and the line leans left. For every substance where the solid is denser, \Delta V is positive and the line leans right. One sign, one observable consequence.
How big is the effect? For ice, L_f = 3.34\times10^5 J/kg, T = 273 K, and \Delta V = -9.1\times10^{-5} m³/kg (ice is about 9 % less dense):
\frac{dP}{dT} = \frac{3.34\times10^{5}}{273\times(-9.1\times10^{-5})} = -1.34\times10^{7}\ \text{Pa/K}
So depressing the melting point by 1 K takes 134 atmospheres. This is worth computing because it demolishes a story that appears in a great many textbooks: that ice skates work by melting the ice under the pressure of the blade. A 70 kg skater on a blade contacting perhaps 1 cm² of ice exerts about 70\times9.8/10^{-4} = 6.9\times10^6 Pa, which is 68 atmospheres, which lowers the melting point by about 0.5 °C. Rinks run at −5 °C or colder. The pressure explanation is off by a factor of ten.
What actually makes ice slippery is a thin liquid-like layer that exists on the ice surface at temperatures well below freezing, because molecules at the surface have fewer neighbours holding them in the lattice — a premelting layer, confirmed by surface-sensitive measurements from the 1990s onwards — together with frictional heating from the blade itself. The Clausius–Clapeyron effect is real and it is simply too small.
Carbon dioxide's missing liquid
CO₂'s triple point sits at 5.1 atmospheres and −56.6 °C. Since atmospheric pressure is 1 atm, which is below the triple point pressure, the line you cross when warming solid CO₂ at room pressure is the sublimation line, not the melting line. Solid carbon dioxide goes straight to gas at −78.5 °C and there is no liquid stage at all — hence "dry ice", and hence its use for chilling things without leaving a puddle. Liquid CO₂ exists perfectly well; it just needs at least 5.1 atm, which is why a CO₂ fire extinguisher sloshes.
The third law
Chapter 3.5 defined entropy changes and never fixed an absolute zero point. Walther Nernst supplied it in 1906:
As the temperature approaches absolute zero, the entropy of a perfect crystal approaches zero.
The counting definition makes this almost obvious. At absolute zero a perfect crystal has every atom in its lowest energy state at its lattice site, and there is exactly one way to arrange that: W = 1. Then S = k_B\ln 1 = 0. Entropy has an absolute zero, not just differences.
This means entropies can be tabulated absolutely, unlike energies, which is why chemistry handbooks list a definite S^\circ for every substance while listing only changes in enthalpy.
The word "perfect" is doing real work. Carbon monoxide, CO, is a nearly symmetric molecule, and when it crystallises each molecule can sit either way round with almost no energy difference. Freeze it fast and the orientations lock in at random, giving 2^N arrangements and a leftover residual entropy of Nk_B\ln 2 = R\ln 2 = 5.76 J/mol/K at absolute zero. Measured: 4.6 J/mol/K, close enough to confirm the mechanism. Ice has residual entropy too, about 3.4 J/mol/K, from the many equivalent ways hydrogen atoms can be placed between oxygen atoms in the lattice — Linus Pauling predicted the number in 1935 by counting, and it was later confirmed.
Why absolute zero is unreachable
The third law has a second, equivalent form that is more useful in the laboratory: no finite sequence of processes can cool anything to absolute zero.
Here is the reason, and it follows from the COP result of Chapter 3.4. A refrigerator's ideal coefficient of performance is T_C/(T_H - T_C). As T_C \to 0, that goes to zero, meaning the work required per joule of heat removed goes to infinity. Cooling from 4 K to 2 K takes far more work than cooling from 300 K to 150 K, despite being a much smaller temperature drop.
More precisely, every practical cooling method works by cycling between two states of the material and exploiting an entropy difference between them. As T \to 0 the third law forces the entropies of all states to converge on the same value, so the entropy difference you are exploiting shrinks to nothing, and each cycle removes less heat than the last. You get closer and closer with more and more effort, and never arrive.
The record is instructive. Liquid helium reaches 4.2 K by boiling. Pumping on it reaches about 1 K. A dilution refrigerator, which exploits the entropy of mixing two helium isotopes, reaches a few millikelvin. Adiabatic nuclear demagnetisation has reached about 100 picokelvin in a spin system, and laser-cooled atomic gases have reached below 100 picokelvin. Each new technique buys a few more orders of magnitude, and zero stays exactly as far away as it ever was.
Where this shows up in your life
A pressure cooker and a vacuum flask are both phase-diagram devices. The pressure cooker moves you up the liquid–gas line so water boils at 120 °C. Freeze-drying moves you down below the triple-point pressure, so ice sublimes directly to vapour without ever melting — which is exactly why freeze-dried coffee keeps its structure and instant coffee dried by heating does not.
Your gas lighter contains a liquid. Butane's critical temperature is 152 °C, comfortably above room temperature, so a modest pressure liquefies it and the clear liquid you can see sloshing in the reservoir is butane under about 2 atmospheres. A propane cylinder is the same idea at higher pressure. But a cylinder of nitrogen or oxygen holds only gas at enormous pressure, because their critical temperatures are −147 °C and −119 °C and no room-temperature pressure will liquefy them.
Sublimation is why snow disappears without melting on a cold sunny day, and why ice cubes shrink and develop a frosted surface after months in a freezer: water molecules leave the ice directly as vapour and redeposit on the coldest surface they find, which is usually the freezer wall.
Rain and cloud seeding depend on metastability. Cloud droplets routinely supercool to −20 °C without freezing, because freezing needs a nucleation site. Silver iodide is used for seeding precisely because its crystal spacing is close to ice's, so it gives water molecules a template to build on.
Supercritical CO₂ decaffeinates your coffee and, increasingly, dry-cleans clothes, because it dissolves organic compounds like a solvent and then leaves without a trace when the pressure drops.
What the next chapter fixes
Everything so far has assumed heat arrives somehow, without asking by what route or how fast. That question matters enormously in practice: it decides how thick a wall needs to be, why a saucepan has a copper base and a plastic handle, and how a star sheds the energy made in its core. Chapter 3.7 works through the three mechanisms — conduction, convection and radiation — and derives the two radiation laws, Stefan–Boltzmann and Wien, well enough to use them on stars in Part 12 and on the Earth's own temperature.